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Theorem rspc2vd 3149
Description: Deduction version of 2-variable restricted specialization, using implicit substitution. Notice that the class  D for the second set variable  y may depend on the first set variable  x. (Contributed by AV, 29-Mar-2021.)
Hypotheses
Ref Expression
rspc2vd.a  |-  ( x  =  A  ->  ( th 
<->  ch ) )
rspc2vd.b  |-  ( y  =  B  ->  ( ch 
<->  ps ) )
rspc2vd.c  |-  ( ph  ->  A  e.  C )
rspc2vd.d  |-  ( (
ph  /\  x  =  A )  ->  D  =  E )
rspc2vd.e  |-  ( ph  ->  B  e.  E )
Assertion
Ref Expression
rspc2vd  |-  ( ph  ->  ( A. x  e.  C  A. y  e.  D  th  ->  ps ) )
Distinct variable groups:    x, A, y   
y, B    x, C    y, D    x, E    ph, x    ch, x    ps, y
Allowed substitution hints:    ph( y)    ps( x)    ch( y)    th( x, y)    B( x)    C( y)    D( x)    E( y)

Proof of Theorem rspc2vd
StepHypRef Expression
1 rspc2vd.e . . 3  |-  ( ph  ->  B  e.  E )
2 rspc2vd.c . . . 4  |-  ( ph  ->  A  e.  C )
3 rspc2vd.d . . . 4  |-  ( (
ph  /\  x  =  A )  ->  D  =  E )
42, 3csbied 3127 . . 3  |-  ( ph  ->  [_ A  /  x ]_ D  =  E
)
51, 4eleqtrrd 2273 . 2  |-  ( ph  ->  B  e.  [_ A  /  x ]_ D )
6 nfcsb1v 3113 . . . . 5  |-  F/_ x [_ A  /  x ]_ D
7 nfv 1539 . . . . 5  |-  F/ x ch
86, 7nfralw 2531 . . . 4  |-  F/ x A. y  e.  [_  A  /  x ]_ D ch
9 csbeq1a 3089 . . . . 5  |-  ( x  =  A  ->  D  =  [_ A  /  x ]_ D )
10 rspc2vd.a . . . . 5  |-  ( x  =  A  ->  ( th 
<->  ch ) )
119, 10raleqbidv 2706 . . . 4  |-  ( x  =  A  ->  ( A. y  e.  D  th 
<-> 
A. y  e.  [_  A  /  x ]_ D ch ) )
128, 11rspc 2858 . . 3  |-  ( A  e.  C  ->  ( A. x  e.  C  A. y  e.  D  th  ->  A. y  e.  [_  A  /  x ]_ D ch ) )
132, 12syl 14 . 2  |-  ( ph  ->  ( A. x  e.  C  A. y  e.  D  th  ->  A. y  e.  [_  A  /  x ]_ D ch ) )
14 rspc2vd.b . . 3  |-  ( y  =  B  ->  ( ch 
<->  ps ) )
1514rspcv 2860 . 2  |-  ( B  e.  [_ A  /  x ]_ D  ->  ( A. y  e.  [_  A  /  x ]_ D ch  ->  ps ) )
165, 13, 15sylsyld 58 1  |-  ( ph  ->  ( A. x  e.  C  A. y  e.  D  th  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1364    e. wcel 2164   A.wral 2472   [_csb 3080
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-v 2762  df-sbc 2986  df-csb 3081
This theorem is referenced by:  insubm  13057
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